Both Marvel and DC Comics have plenty of unique weaknesses for their respective superheroes, but the style of the weaknesses drastically changes between companies - and DC's method of limiting their heroes suggests they cannot move on from the Silver Age of Comics, no matter how hard the try. With both companies using incredibly overpowered heroes like Captain Marvel and of course Superman, it is only natural for writers to impose strict limits on their abilities and find ways to cancel them out altogether. But in general, DC uses external weaknesses while Marvel utilizes internal weaknesses - and therein lies DC's problem.
The 1940s gave rise to the Golden Age of Comics, in which there were no established "rules" for superheroes to follow. As such, many were astoundingly overpowered, such as Alan Scott's Green Lantern. Writers quickly realized that a hero who could accomplish anything would make for rather boring stories, so Alan Scott's Green Lantern ring was unable to affect any material made out of wood. DC editors didn't know it at the time, but this wold set an interesting precedent for their heroes in the future.
The hero Shazam would revert to Billy Batson immediately if he ever spoke his own name. Aquaman would begin to grow weak after an hour away from water (namely the salt water of the ocean). Hal Jordan's Green Lantern was unable to affect any yellow-colored object, Wonder Woman lost her powers if her wrists were even bound by a man, and the Martian Manhunter has a vulnerability to fire (which is occasionally psychosomatic in some interpretations. The best-known superhero weakness is Superman's aversion to Kryptonite; he grows weak when the substance is near, and prolonged exposure will result in death. These are all external weaknesses, in which the superhero loses their powers thanks to an element that doesn't originate from a character's powers or personality. Perhaps sensing a trend, the editors at Marvel tried a different strategy.
Marvel character's weaknesses often stem from their own personality and character flaws. While Tony Stark's arc reactor is a physical weakness, his real weakness is his own ego, hubris and self-destructive attitude. Spider-Man's own guilt and selfish behavior often lead to disaster (see the conclusion of Spider-Man: One More Day for a prime example). Captain Marvel may be the companies' Superman analogue, but her anger, a might-makes-right approach and an inability to take perspective are major weaknesses (all of which are demonstrated in Civil War II).
Occasionally, an external weakness might trigger an internal character flaw (the Thing's weakness is not exotic rocks from space but his own self-esteem issues exacerbated by his inability to revert to human form, for example). DC has noticed the effectiveness of Marvel's weaknesses and have attempted to erase their own; Aquaman no longer needs to be close to water and Wonder Woman's Golden Age weakness has been removed entirely. But the fact remains that Marvel's weaknesses serve to make their characters human - even if they themselves are not - while DC Comics would rather have characters with exotic weaknesses that put even more distance between the reader and the superhero.